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talk012898: Informal Galaxy Seminar V Wonho Oh Department of Applied Mathematics and Statistics State University of New York at Stony Brook An Introduction to MPI-2 MPI is a new library of specification for message-passing, proposed as a standard. The version 1.1 of MPI was recently extended to a new standard MPI-2. The extensions include process creation and management, one-sided communications, extended collective operations, external interfaces, I/O, and some miscellaneous topics. On Galaxy, the MPICH implementation is based on MPI-1.1. Any impelmentation based on MPI-2 is not known to the speaker yet. In this talk, basics of MPI-1.1 and some extensions in MPI-2 will be explained.
talk021398: Friday, 02/13/98 (Same time.) Snezhana I.Abarzhi Department of Mathematics, University of North Carolina-Chapel Hill The Bubble Problem in the Rayleigh-Taylor Instability. We study theoretically the "bubble problem", a highly non-linear stage of the Raylegh-Taylor instability. For steady solutions family, the local stability criteria are established in 3D and 2D cases, and the problem of uniqueness of RTI steady bubble is decided. We find the dependence of stable solutions on 3D flow symmetry, and show that dimensional crossover in RTI is discontinuous. Global flow instabilities due to sub-harmonic modulations are analyzed in terms of symmetry theory. Agreement with existing experimental and numerical data is well.
talk030298: Monday, 03/09/98 (Time: 10; Place: P131) Steven Liebling Center for Relativity, Department of Physics The University of Texas at Austin Austin, Texas Title: Modeling the Harmonic Map: Black Holes, Singularities, and Topological Defects Abstract: Studies of black hole formation in the harmonic map model in spherical symmetry have led to interesting results. The model contains a dimensionless constant, $\kappa$, which parameterizes the curvature of the internal space. For $\kappa$ very negative and positive, evolutions suggest the formation of finite-time singularities occuring with or without gravity. In the former case, these singularities appear to be simply a cooridinate singularity, while in the latter, it seems likely the singularity is coordinate invariant. To further study the regime $\kappa<0$, I have constructed a flat space (no gravity), three-dimensional model of a triplet scalar field with a symmetry-breaking Mexican-Hat potential, which, in the limit of inifinite coupling, corresponds to the harmonic map. Here, I study the evolution of textures and monopoles, and, in particular, clarify the type of textures whose collapse nucleates monopole-antimonopole pairs.
talk030998: Monday, 03/09/98 (Same time: 10:30) Scott Wunsch Department of Mathematics University of Chicago Anomalous Scaling in Passive Scalars: A Simple Advection-Diffusion Model Abstract: It is known experimentally that tracer particles (passive scalars) at high Peclet number advected by a high Reynolds number flow exhibit anomalous (non-gaussian) scaling in their structure functions. This was once attributed to the anomalous scaling of the underlying velocity field; however, recent theoretical ideas suggest that the scaling is inherent in the passive scalar advection-diffusion equation. This talk will review these developments and then discuss a simple numerical model which has been used to test some of these new ideas.
talk032698: Approximation of Contaminant Transport through Porous Media Anna Maria Spagnuolo Center for Applied Mathematics Purdue University A differential system describing the flow of a multi-component, nuclear fluid in a reservoir will be presented. A linear partial differential equation models the concentration of each descendent encountered in the decay chain in this miscible displacement. From conservation of mass for the mixture, an elliptic equation for pressure is derived. Conservation for the displacing fluid leads to a convection-dominated parabolic equation for the concentration of each contaminant, which is coupled to the pressure equation. The pressure and Darcy velocity of the mixture are approximated by a mixed finite element method. A new numerical scheme that conserves mass, the Modified Method of Characteristics with Adjusted Advection (MMOCAA), will be introduced for approximating the concentration of each contaminant in the chain. Convergence of the numerical solution has been established.
talk032798: Friday, 03/27/98 (Same time: 10:30) Adam Halasz Department of Physics University of Stony Brook Applications of Random Matrices in QCD Lattice simulations are possibly the most reliable tool in investigating Quantum Chromodynamics (QCD), the field theory that describes the strong interaction. We discuss results based on a schematic model of QCD which uses random matrices, i.e. matrices with random elements. This model reproduces universal features of QCD, which are responsible for certain nontrivial effects of the finite lattice size. It is also cheap to simulate, allowing us to study computational difficulties like those encountered in the case when the QCD action is complex.
talk041598: Wonho Oh Department of Applied Mathematics and Statistics State University of New York at Stony Brook Applications of Kriging to Conditional Simulation and Image Segmentation Kriging is a statistical interpolation method which was developed in geostatistics. It gives an unbiased estimator with minimum error variance. It has been used to generate realizations of random fields conditioned to (true) measured data. Recently, we have developed an application of kriging to image segmentation. It utilizes the spatial covariance (second order moment) of the image. In this talk, basic properties of Kriging will be introduced and its application to image segmentation will be discussed. Some results on synthetic and real images of rock samples will be presented.
talk042298: Informal Graduate Student Gathering: An introduction to neural networks Cheng-Hung Chou Department of Applied Mathematics and Statistics University at Stony Brook Neural networks, or artificial neural networks, are an emerging technology rooted from many disciplines, e.g., engineering, physics, and neuroscience. The brain is a highly complex, nonlinear, and parallel information processing system. It has the capability of organizing neurons to perform certain computations (e.g., pattern recognition, classification) many times faster than the fastest digital computer in existence today. The design of a neural network is motivated by analogy with the brain. Neurobiologists look neural networks as a research tool for the interpretation of neurobiological phenomena. While engineers look it as a new way to solve problems hard to solve by conventional way. This talk will give introduction about some basic ideas of neural networks and their applications.
talk042898: Note the time and date for this talk: 4/28/98 at 11AM. in Physics P-119. Richard Holmes Los Alamos National Laboratory Thermonuclear Applications Group Los Alamos, NM 87545 RAGE: A Continuous Adaptive Mesh Refinement (CAMR) Code In this talk I will describe the code RAGE (RAGE stands for Radiation Adaptive Grid Eulerian) being developed at Los Alamos in collaboration with SAIC. RAGE is a continuous adaptive mesh refinement code (CAMR), meaning that every cell in a problem is evaluated for subdivision at every timestep. While it has radiation transport capabilities, I will concentrate solely on the hydrodynamic aspects of the code. I will briefly describe the code and its fundamental algorithms. I will then spend most of the time showing results from simulations in 2D and 3D, concentrating on comparisons to other codes (including FronTier) and to experiments.
talk052098: Grafton Hui Department of Mathematics HKUST A unified Coordinate system for Computational Fluid Dynamics
talk092398: Informal Galaxy Seminar VI Kevin Chen HS Instruments, Inc. kchen@aestheticism.com What can we do with Windows NT 4.0? Windows NT 4.0 is the most recent incarnation of Microsofts New Technology (NT) line of graphical operating system. This operating system is aimed at running applications at the corporate and professional level with "mission critical" requirements. These requirements have made Windows NT into a complex system to maintain and program. Through an overview of NT's design goals and application development tools, the aim is to cast these complexities as a list of capabilities that one selects to get a job done. Topics will cover: A Brief History of Windows; Windows NT Design Goals Compatibility, Scalability, Portability, Security Distributed Processing Reliability and Robustness Localization Extensibility Overview of Windows NT 4.0 NT 4.0 Architecture Graphics Device Interface Supported File Systems Security Model and The System Registry Networking Capabilities Maintenance Programming in Windows NT 4.0 Application Programming Interface (API) from Win16 to Win64 Executables and Dynamic Linking Libraries (DLLs) Component Object Model (COM) Distributed Component Object Model (DCOM) Visual C++ and Visual Basic Other Application Development Services Data Access Object (DAO) OpenGL and DirectX APIs Internet Explorer Object DCOM and Distributed Computing Cryptography An Example of Windows NT 4.0 in Manufacturing Process Control
talk101498: Alex Lipton Bankerstrust co. Linear and Nonlinear Pricing Problems in Mathematical Finance In the present talk we discuss several interesting and important problems of mathematical finance and show how they can be solved in the framework of stochastic calculus and partial differential equation theory.
talk102898: Fixed Income Research Needs on Wall Street. Joe Langsam Morgan Stanley Dean Witter & Co. Wednesday Oct. 28, Rm 1-122A Math. Building 12:30-1:30 p.m. Joseph Langsam is a managing director of Morgan Stanley Dean Witter and is responsible for the analytical research activities for the Fixed Income Division. A graduate of MIT, Mr. Langsam earned a Ph.D. in Urban Studies and Economics from MIT and a Ph.D. in Mathematics from the University of Michigan. Mr. Langsam joined Morgan Stanley &Co. Incorporated in 1985, was named Vice President in 1987, Principal in 1993, and Managing Director in 1997. Mr. Langsam is a member of the firm Risk Management Advisory Board and chairs the methodology committee for the Board. Mr. Langsam serves on the Advisory Board for the Department of Mathematics of Carnegie Mellon University. Dr. Langsam will talk on the research needs of the Wall Street. An alternative title of his talk is "A research wish list: Research needs at the bank level." Note: This talk will started at 12:30AM
talk110498: Xiaolin Li AMS/Stony Brook Three Dimensional Front Tracking We present a simplification of the Front Tracking method for the study of fluid interface instabilities in three dimensions. This includes a simplified algorithm for the resolution of interface geometry and a partially tracked algorithm for the shock-contact interaction. These algorithms will greatly enhance the capability and robustness of the Front Tracking as an active tracking method. Two problems have been solved, they are: (1) interface reconstruction at the level of rectangular block, a method based on an analysis of micro-topology and block isomorphism and (2) a non-oscillatory shock-contact interaction with untracked shock and tracked contact surface. The third problem involves a partially adaptive oblique solver, its implementation will cure the secondary instability at the shock transition through a contact surface in two and three dimensions.
talk111198: Chi-Wang Shu Division of Applied Mathematics Brown University Note: This talk will started at 11:30AM and may last for 1.5 hours. Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Conservation Laws He will present the basic ideas and recent development in the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for conservation laws and their applications to computational fluid dynamics. ENO and WENO schemes are high order accurate finite difference or finite volume schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in computational fluid dynamics and other applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.
talk111898: Similarity Theory and Inertial Range Scalings in Fluid Turbulence Shiyi Chen Center for Nonlinear Studies Los Alamos National Laboratory Theories in the inertial range scaling of fluid turbulence, such as Kolmogorov's similarity theory, are largely based on the understanding of the turbulent dissipation function. In this talk, we report our recent studies of the inertial range scalings of energy dissipation rate and enstrophy (vorticity squared) using high resolution direct numerical simulation at moderate Reynolds numbers for homogeneous and isotropic turbulence. It is found that the enstrophy is more intermittent than dissipation and enstrophy and dissipation have different scaling exponents, consistent with one-dimensional surrogate experimental measurements at high Reynolds numbers. Based on the empirical data, we further argue that Kolmogorov's refined similarity hypothesis (RSH) needs to be modified for transverse velocity increments, and propose an alternative. In this new form, transverse velocity increments bear the same relation to locally averaged enstrophy (squared vorticity) as longitudinal velocity increments bear in RSH to locally averaged dissipation. We support this hypothesis by analyzing high-resolution numerical simulation data for isotropic turbulence. RSH and its proposed modification for transverse velocity increments (RSHT) appear to represent two independent scaling groups at finite Reynolds numbers. The physical interpretation related to these similarity solutions will be offered. Implications of this new similarity on numerical modeling of fluid turbulence will be discussed. Upon closing, the inertial range scalings in the limit of very high Reynolds numbers will be discussed based on axisymmetric vortex models.
talk120298: Modeling and Computation of Interfacial Averages in Two-Phase Flow Dave Saltz AMS/Stony Brook In two-phase flow modeling, the correct representation of volume, momentum, and energy exchange across material interfaces is of primary importance. By their very nature, these terms largely account for the structure of the two-phase flow, and hence for the differences among two-phase flow models. In this talk, we will describe two distinct approaches to modeling interfacial effects in two-phase flow. Such interactions can be formally related to averages of physical quantities at material interfaces, thereby permitting their computation from direct numerical simulation data. The front-tracking code FronTier is well suited for this purpose, as it has the important advantage of accurate resolution of fluid states at material boundaries. We will describe the use of FronTier to simulate fluid mixing by Rayleigh-Taylor instability, and discuss the problem of extracting meaningful interfacial averages from the numerical data, as a means of testing recently proposed models.
talk120898: AMS Workshop to honor visitors from the Beijing Institute of Applied Physics and Computational Mathematics 9:00AM--1:40PM December 8, 1998 AMS Seminar Room 1-122 Coffee and Free Lunch will be served 9:00-- 9:05 James Glimm Welcome 9:05-- 9:45 James Glimm Scientific Computing 9:45--10:25 Beijing Visitors Research Programs 10:25--10:35 Break 10:35--10:55 Brent Lindquist Stochastic Geometries 10:55--11:15 Yuefan Deng Molecular Dynamics 11:15--11:35 David Saltz Fluid Mixing 11:35--11:55 Brad Plohr PDEs in Continuum Mechanics 11:55--12:00 Break 12:00-- 1:00 Lunch (Served in Conference Room) 1:00-- 1:20 Xiaolin Li Fluid Instabilities 1:20-- 1:40 Joe Mitchell Computational Geometry Participating visitors from the Beijing Institute are: Prof. Shen Longjun, Deputy Director Beijing International Center for Computational Physics (BICCP) Institute of Applied Physics and Computational Mathematics (IAPCM) Prof. He Xiantu, Academician of China Academy of Sciences Prof. Zhang Tianyuan, Director, Office of Foreign Affairs, IAPCM; Office of BICCP All members of College of Engineering and Applied Sciences are welcome. No Registration is Needed. For convenience of planning, please send an email to Professor Yuefan Deng at deng@ams.sunysb.edu if you like to attend.
talk121698: Domain Decomposition Methods for Subsurface and Surface Flow Problems Mary F. Wheeler Department of Aerospace Engineering & Engineering Mechanics, Department of Petroleum Engineering, and Center for Subsurface Modeling, TICAM University of Texas at Austin Fluid flows at and below the earth's surface are the cause and cure for problems of water and soil pollution. Petroleum and natural gas production depends on flows in the earth's subsurface. Length scales of practical and economic interest range from tens of meters to kilometers. Moreover, different physical processes occur simultaneously in different parts of the domain (e.g. single flow within an aquifer, multiphase flow in the vadose zone above the aquifer, and shallow water transport in a river or wetland in contact with the porous media). Our basic approach for parallel multiphysics/multiscale simulation uses the concept of multiple blocks or domains. In this approach, two levels of domain decomposition for parallel computation are considered: physical and computational. First the physical problem is decomposed with appropriate hierarchical models (representing the geometry, geology, and chemistry/physics/biology); then the computations must be efficiently decomposed on a parallel machine. We emphasize a multiblock or macro-hybrid approach to decomposition, in which we describe a domain as a union of regions or blocks. This offers great flexibility to accomodate the shape of the external boundary, the presence of internal features such as faults, the need to refine a region of the domain (and thus to treat it as a distinct block), and to accomodate models of multiscale and multiphyscial phenomena. Specially chosen mortar spaces are introduced for the primary unknowns along the domain interfaces in order to provide numerical models for multiblock domains which are consistent with the physical and engineering description of the underlying equations. That is, the equations hold with their usual meaning on the subdomains, which have physically meaningful interface boundary conditions between them. Numerical algorithms for subsurface and surface flow problems which illustrate these multiblock domain decomposition concepts will be described. Parallel computational results will also be presented.
talk012799: "Discontinuous Galerkin Finite Element Methods for Hamilton-Jacobi Equations" Changqing Hu Brown University In this talk, I will present discontinuous Galerkin finite element methods for solving the nonlinear Hamilton-Jacobi equations. The methods are based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. A least-square procedure is used to keep the consistency for the approximation. The methods have the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high order accuracy with a local, compact stencil, and are suited for efficient parallel implementation. One and two dimensional numerical examples will be given to illustrate the capability of the methods.
talk021099: Effects of small surface tension in free surface flow Michael Siegel Department of Mathematics New Jersey Institute of Technology The influence of small surface tension in some time evolving free surface flow problems will be considered. Attention will be focused on flows for which, in their simplest formulation (i.e., neglecting surface tension or other regularization) the interface is known to develop cusps, corners or other singularities. This situation presents some interesting challenges for numerical computation. In the first part of the talk, the effect of arbitrarily small surface tension on a time evolving interface in Hele-Shaw flow (flow between two narrowly spaced glass plates) will be considered. Numerical and analytical evidence is presented to show that, as surface tension tends to zero, the motion of the interface can deviate significantly from the zero surface tension motion in O(1) time (i.e., after a time interval that is independent of surface tension). This happens even when the zero surface tension solution is smooth. In the second part of the talk, some interesting effects of non-constant surface tension in free surface Stokes flow will be discussed.
talk020599: Informal Gathering Friday (2/5/99) 10:30AM in AMS Seminar Room Gene Chips and Gene Expression Networks Tim Ting Chen, Ph.D. Harvard Medical School Room 407, HIHG/HIM 77 Avenue Louis Pasteur Boston, MA 02115 USA Gene chip, a new technology, is having a great impact in the discovery of biology and medicine. Today, we are able to profile all the genes expressed in a cell by using this chip. However, how to deal with the massive data generated every day is the bottleneck for most applications. In this talk, we will introduce this technology and talk about our data analysis methods. No biology background is needed.
talk021799: Decay Rates of Internal Waves in Viscous Near-Critical Fluids Katherine Gurski Department of Mathematics University of Maryland Near the liquid-vapor critical point, a fluid exhibits large compressibility, creating a variation of density as a function of depth. Such "density-stratified" fluids are able to support internal gravity waves. Operator-theoretic methods are developed which predict the existence of internal waves with arbitrarily small damping rates in the presence of viscosity which correspond to one mode of an overdamped oscillator. Asymptotic analysis and numerical calculations find a large increase in the damping rate as the temperature of the fluid approaches the critical temperature.
talk022299: Monday, Feb 22, 10.30 am seminar room: Yong Jung Kim Department of Mathematics, Madison Wisconsin Topic 1: A SELF-SIMILAR VISCOSITY APPROACH FOR THE RIEMANN PROBLEM IN ISENTROPIC GAS DYNAMICS Abstract: We study the Riemann problem for the system of conservation laws of one dimensional isentropic gas dynamics in Eulerian coordinates. We construct solutions of the Riemann problem by the method of self-similar zero-viscosity limits, where the self-similar viscosity only appears in the equation for the conservation of momentum. Topic 2: INVARIANCE PROPERTY AND THE BURGERS EQUATION Abstract: We can observe invariance property in many examples. For example the viscous Burgers equation is invariant under the group of substitutions $x\to cx, t\to c^2t, u\to u/c$. This kind of invariance property characterizes the behavior of the solutions. The purpose of this work is to develop a method to study differential equations under the invariance property. We study the viscous Burgers equation in terms of new variables which are invariant under those substitutions. The Burgers equation is transformed into a conservation law with space dependent flux under these variables. This study provides us another explanation of the well known behavior of the Burgers equation. It also provides us a good understanding of the evolution of the solutions, which was not possible from the original equation. This approach make it possible to see the long time behavior numerically, which is impossible with the original equation.
talk031099: Prof. Mark Knackstedt Dept. Applied Math. Australian National University Canberra, Australia Studies of Multiphase Flow Abstract: A detailed understanding of multiphase flow phenomenon in porous media at different scales is crucial to the prediction of oil and gas recovery, geothermal energy extraction and groundwater pollution abatement in underground reservoirs. In petroleum engineering for example, reservoir simulators are routinely used by the petroleum industry to predict production potential of oil and gas fields. The accuracy of these predictions depend on capillary pressure and relative permeability on the scale of a reservoir grid block (~1000 cu. m). These parameters cannot be determined at this scale and are instead measured in laboratory experiments on small plugs cut from reservoir core (~ 10 cc). A typical plug represents a minute fraction of the total grid block volume. A major problem in performing realistic field simulations is relating laboratory core measurements to the grid-block scale. In particular relative permeability is strongly influenced by heterogeneity and heterogeneity may occur at all scales in the reservoir from pore-scale to field-scale. To gain a fundamental understanding of multiphase fluid-flow properties of porous media requires an intergrated approach in which three problems must be simultaneously addressed: Accurate characterisation of the pore geometry and heterogeneity Accurate modelling of multi-phase flows at the pore scale Ability to solve flow problems on massively parallel grids. We discuss progress in integrating these three areas into one program and subsequent improvements in prediction of multi-phase flow properties of porous media. Experimentally we have used X-ray CT and SEM to measure pore morphology and the spatial distribution of porosity in rock cores. A precise description of the distribution and pore-scale mechanisms for multi-phase flow is also presented. This provides a basis for more accurate modeling. Finally, we have developed a new algorithm for network invasion simulations of multi-phase flow in porous media that are orders of magnitude more rapid than traditional algorithms. This has enabled us to directly simulate multi-phase flow properties in three dimensions on grids of up to a billion pores. We discuss implications of the work to core-scale measurements and upscaling.
talk0310A99: Wednesday, March 10, 1:00-2:00 A Weak Turbulence Model for the 1-D Dispersive Wave Equations Chongsheng Cao University of California, Irvine The principle of the weak turbulence is that the nonlinear interaction is smaller than the linear effects caused by the dispersion. Based on this principle and the fact that energy transfer takes place through resonant wave interactions, the closures of the statistical kinetic equations have been developed. The numerical simulations show that the weak turbulence theories are excellent in approximating the statistical stationary solutions of the dispersive wave equations.
talk031799: Roman Samulyak Department of Mathematical Sciences New Jersey Institute of Technology Newark, NJ 07102-1982 phone: 973-596-8391 fax: 973-596-6467 E-mail: rosamu@eclipse.njit.edu DYNAMICAL SYSTEMS ASSOCIATED WITH PARTICLE FLOW MODELS: THEORY AND NUMERICAL METHODS Abstract A new class of integro - partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian equations of motion of a many-particle system incorporating widely used inelastic particle-particle force formulas. By using Taylor series expansions, these models can be approximated by a system of partial differential equations of the Navier-Stokes type. A numerical algorithm based on the idea of higher and lower modes separation in the theory of approximate inertial manifolds for dissipative evolutionary equations is developed in a finite-difference framework. The method is applied to the granular flow dynamical system. Numerical calculations show that this method has several advantages compared to standard finite-difference schemes. Some analytical and numerical solutions to the dynamical system describing granular flows in vibrating beds are presented. We found that even in the simplest case where we neglect the arching phenomena and surface waves, these solutions exhibit some of the typical features that have been observed in simulation and experimental studies of vibrating beds. The approximate analytical solutions to the governing system of equations were found to share several importent features with actual granular flows. Using this approach we showed the existence of the typical dynamical structures of chaotic motion. By employing Melnikov theory the bifurcation parameter values were estimated analytically. The vortex solutions we obtained for the perturbed motion and the solutions corresponding to the vortex disintegration agree qualitatively with the dynamics obtained numerically. A class of one-dimensional models describing the dynamics of thin granular layers and some related problems of fluid mechanics were studied numerically and analytically. We proved the Liouville-Lax integrability of some of these models. By proving the exact integrability of the systems, the quasi-periodicity of the solutions was explained as well as the observed regularity of the numerical solutions.
talk032299: 1:30PM in Seminar Room Instabilities and formation of complex domain patterns in reaction-diffusion systems Cyrill Muratov Courant Institute New York University Reaction-diffusion systems of activator-inhibitor type exhibit remarkable diversity of spatio-temporal patterns. In certain limits these patterns can be described by their geometric characteristics becuase of the presence of sharp interfaces and strong separation of length scales. I will present asymptotic theory of instabilities of such patterns and perform complete stability analysis for the simplest localized patterns in 2 and 3 dimensions. The obtained instability criteria turn out to be universal and practically independent of the concrete nonlinearities of the system. I will use these results to explain pattern formation scenarios observed in the numerical simulations of a concrete reaction-diffusion model.
talk033199: Some Analysis for Moving Mesh Galerkin Method Yingjie Liu Department of Mathematics The University of Chicago This work tries to increase our understanding of why moving mesh methods often work very well. We first start with restricted Miller's method for a simple heat equation to show how mesh movement can model diffusion. Then we study moving mesh Galerkin Method for advection-diffusion equations to obtain some error estimates which are independent of the bound of advection. Symmetric error estimates with different norms may provide insight for balancing the modelling of diffusion and advection by mesh movements. If time allows, I will briefly talk about some results on moving mesh mixed method, or for fun, the tangential suspension system.
talk040799: A Method for the Spatio-temporal Registration of the Expression Patterns of Drosophila Segmentation Genes Dr. Ekaterina Myasnikova Institute for High Performance Computing and Data Bases P.O.Box 71, St.Petersburg, 194291 Russia Observational information about gene expression patterns is becoming available in unprecedented amounts. Among the genes being studied are some which form reasonably small networks of five to twenty genes which control well defined aspects of fundamental morphogenetic processes. Simultaneous observation of the expression patterns of all of the genes in such a network over time at cellular resolution is of great biological importance. A twofold difficulty in obtaining these data must first be overcome, however. First, the data is obtained from fixed tissue with the consequence that temporal dynamics must be reconstructed from many samples, each obtained at a separate time. Second, current technology permits the observation of only two or three gene products simultaneously from an embryo. This means that a spatial expression map of all the components of a network must be synthesized from many separate observations of a few components, each made on a separate embryo. These two issues constitute the "registration problem" for gene expression data in time and space. Here we present new results that address both issues in the context of a particular biological system, such as a segmentation gene network in the embryo of fruit fly Drosophila melanogaster. Expression of segmentation genes is largely a function of position on the anterior-posterior (A-P) axis, and so can be well represented in one dimension. Gene expression is monitored by confocal scanning of fixed embryos. Images of the patterns are processed so that each nucleus in the processed dataset is labeled numerically and specified by the following features: x and y coordinates of its centroid together with quantitative values for the average intensities of gene expression for up to three genes over each nucleus. We describe how data of this type can be combined into an integrated atlas. The creation of such a dataset is accomplished through the application of image registration techniques in both space and time. The simplest 1-D registration method is based on the quadratic spline approximation of the expression patterns. This approach allows to identify the most essential features of each pattern and to express them in terms of a certain number of spline parameters. The temporal classification of the patterns requires the approximation method which provides the better fit to the experimental data than the regular spline-based one. We apply the modified quadratic spline approximation, for which the condition of continuous first derivatives at knots is not imposed, keeping only the continuity requirement for the function itself. Such modification provides a better approximation of sharp peaks on the experimental curves. These methods will allow us to construct a map of all relevant expression domains from a series of embryos of the same or different age.
talk041499: Charged particle diffusion in collisionless turbulent plasmas in space Dr. Miriam Forman, Department of Physics and Astronomy The coupling of cosmic rays and other energetic particles to their space plasma environment, through turbulent magnetic fields (rather than particle-particle interactions), is a key process in many important phenomena in space physics and astrophysics. Among these are: the origin and acceleration of cosmic rays in supernova shock waves; solar flares and the 22-year solar modulation of galactic cosmic rays; and particle transport and acceleration in other interesting places in the universe, such as gamma-ray burst sources. The first object of cosmic-ray diffusion theory is to suitably average the true magnetic forces exerted on particles by the typical turbulent magnetic fields in space, into a scattering operator in a transport equation for the average distribution function. Second, is to solve the resulting integro-differential equation, to determine the shape of the average distribution function in momentum space in response to a spatial gradient. Then we can evaluate the macroscopic transport coefficients, and solve macroscopic transport problems such as the origin and flow of cosmic rays in the Universe. This procedure is analogous to many other transport theories in chemistry, physics and applied mathematics, in which a suitable average over the fundamental physics of the interactions of individual particles with their environment, or each other, is used to derive a macroscopic transport equation for a large number of particles. There are important differences, but the similarities are worth exploring. We might learn useful approaches or methods from other types of transport theory. This talk will review the role of diffusive transport and acceleration of energetic particles in astrophysics, and methods used to derive the scattering operator in turbulent magnetic fields and solve the scattering equation.
talk041999: ******Special seminar at 4PM on Monday (4/19/99) in AMS seminar room****** Joseph W. Haus RPI Physics Department The Science of Photonic Band Structures A general overview of selected concepts for periodic dielectric structures (i.e. photonic band structures) is given. The group theoretic analysis of mode symmetry is discussed with potential applications, especially, the photonic crystal laser. Experimental results on the laser and their analysis are provided.
talk050599: Dr. Len Brin Department of Applied Mathematics and Statistics University at Stony Brook AUTOMATIC MESH REFINEMENT Automatic (adaptive) mesh refinement is a method used to generate a nonuniform grid during numerical calculations, concentrating cells (and therefore compute time) around the most sensitive regions of the calculation. Two distinct methods have been developed: 1) (Berger's) patch based AMR 2) tree based AMR Current work is being done to develop tree based AMR code in FronTier. The most obvious (side) effect of this effort on FT is the "conversion" of FronTier from C into C++. I will discuss the code development, current status of the project, and how it may affect anyone else working on FT.
talk052799: ****Note: This seminar will take place at 10:30 Thursday in Seminar Room**** S.Abarzhi Department of Applied Mathematics and Statistics University at Stony Brook Bubbles in Richtmeyer-Meshkov instability We study theoretically the evolution of the Richtmeyer-Meshkov instability for 2D and 3D flows. We integrate equations explicitly from initial to advanced stage of the instability and evaluate the influence of initial conditions. For 3D highly-symmetric flows theoretical solutions have a universal form when expressed in dimensionless units. At a fixed length-scale the values of the bubble velocity depend strongly on the 3D flow symmetry, 2D flows are slower than 3D ones. We discuss the influence of initial data on the instability evolution.
talk071499: Particle Accelerator Numerical Simulation Isses Dr. Alfredo U. Luccio Brookhaven National Laboratory Computer simulation is a very important tool for the study of particle accelerator beam dynamics. The beam in the accelerator is modeled as an ensemble of representative macroparticles, and algorithms of PIC (particle-in-Cell) propagation are used. The main purposes of the simulation are (i) design new accelerator structures, (ii) understand and optimize existing accelerators, (iii) do model-based accelerator control. Parallel computing is the only way to address and solve many of the problems, when very intense beams are present and the configuration of the boundary (acceleration chamber walls) is complicated.
talk091699: Note: this is a joint Math/AMS colloquium and will be held at 4:00 pm in Room P-131. Divergence-Measure Vector Fields and Hyperbolic Conservation Laws Professor Gui-Qiang Chen Northwestern University In this talk, we first discuss a relation between divergence-measure vector fields, a class of vector fields in $L^p$, and nonlinear hyperbolic conservation laws. This class of vector fields includes, in particular, a class of vector fields with bounded variation; however, there are essential differences. We analyze divergence-measure vector fields and present their mathematical theory, including the Gauss-Green formula, normal traces over Lipschitz surfaces, etc. Then we discuss their applications to solve several important problems for nonlinear hyperbolic partial differential equations in conservation form and related topics.
talk092999: Multiscale Finite Element Computations for Flow and Transport in Strongly Heterogeneous Porous Media Professor Thomas Y. Hou Applied Math Department California Institute of Technology Abstract We introduce a multiscale finite element method for computing flow and transport in strongly heterogeneous porous media which contain many spatial scales. The method is designed to effectly capture the large scale behavior of the solution without resolving all the small scale features. This is accomplished by constructing the multiscale finite element base functions that incorporate local microstructures of the differential operator. Our method is applicable to general multiple-scale problems without restrictive assumptions on scale separation and periodicity. Convergence of our method has been established in the case of periodic oscillatory structures. The rate of convergence is shown to be independent of the small scales of the solution. We demonstrate the accuracy and robustness of our method through extensive numerical experiments, which include the scale-up of two-phase flows with strongly shear random permeability, wave propagation through heterogeneous media, and convection enhanced diffusion. Steady conduction through fiber composites and flows through random media with normal and fractal porosity distributions will also be considered. Parallel implementation and performance of the method will be addressed.
talk102299: Solving the Inverse Problem in Drosophila by Means of Serial and Parallel Simulated Annealing. Prof. John Reinitz Mount Sinai Medical School *** SPECIAL TIME: 12:30PM *** Friday October 22 ABSTRACT Simulated annealing is a global optimization method inspired by statistical mechanics. Parallel implementation of this method is an open research problem. This talk will describe progress in the development of new parallel simulated annealing algorithms. This work is part of a long term project devoted to solving fundamental problems in animal development. We model the segment determination process in Drosophila by large systems of nonlinear ordinary differential equations (ODE's). Parameters occurring in these equations must be determined from time series data by solving the inverse problem. Here the inverse problem is posed as a large scale optimization problem which is solved by simulated annealing. We expect that the results described will be applicable to a wide variety of optimization problems, both inside and outside of biology.
talk102799: N-body calculations with MD-GRAPE Bruce Elmegreen IBM Watson Research Center Yorktown Heights, NY 10598 ABSTRACT The history of the development of the GRAPE family of accelerators will be discussed, with emphasis on the molecular-dynamics chip, MD-GRAPE. The purpose of this chip is to accelerate the calculation of arbitrary forces between all pairs of bodies in an N-body calculation. It is useful for molecular dynamics, fluid dynamics, plasma dynamics, magnetics, and astrophysics. Results from the first version, made in 1995, will be reviewed, and progress on the second chip, MD-GRAPE2, will be summarized. Specifications of MD-GRAPE2 speed and memory will be given, along with some suggestions about how it might be used.
talk121399: AN OVERVIEW OF RAYLEIGH-TAYLOR MIXING RESEARCH AT TEXAS A&M Malcolm J. Andrews Department of Mechanical Engineering Texas A&M University College Station, TX 77845-3123 Abstract Rayleigh-Taylor mixing occurs when a heavy fluid is placed over a light one under the influence of gravity. This unstable buoyancy configuration drives small scale perturbations at the interface to grow and interact forming a turbulent mix region. Such mixing occurs in the atmosphere, oceans, rotating machinery, swirling flow in heat exchanger tubes, and other more exotic environments such as Inertially Confined Fusion. The presentation will discuss past and current research on Rayleigh-Taylor mixing, and include experimental, computational, and theoretical work. Recent experiments suggest the growth-rate parameter alpha may not take a self- similar universal value in the experiments, possible reasons for this observation will be discussed.
talk020200: Mathematical modeling and numerical simulation of star formation Christian Klingenberg Wuerzburg University presently visiting JPL and Applied Math California Institute of Technology Abstract We consider astrophysical jet flow associated with star formation. This is modeled by a system of conservation laws. The non-linear nature of these differential equations requires numerical discretisation in so-called conservation form. For our astrophysical problem this puts us into a quandary: internal variables like pressure and temperature can no longer be computed accurately. We propose the following way out: Embed the astrophysical jet model into a more complete model. There it is readily possible to compute the internal variables. Then we project these variables back to the original model. We can prove that the translation of this into a numerical procedure leads to reliable solutions. I will try to make this talk accessible to graduate students. It will be illustrated with pictures showing actual astronomical observations and numerical simulations.
talk020900: The Mathematical Theory of Three-Phase Flow and Wave Structure in Enhanced Oil Recovery D. Marchesin, Instituto de Matematica Pura e Aplicada, Brazil High-resolution simulations with negligible numerical diffusion have been employed to understand the flow of three immiscible fluid phases in a porous medium. We have studied flow of idealized oil, gas, and water in a long horizontal thin core, initially containing a high oil saturation, when water and gas are injected alternately (the Water-Alternating-Gas, or WAG, enhanced oil recovery strategy). The simulations indicate three important features in the flow. Closest to the production end is an oil bank, across which the oil saturation decreases substantially. Surprisingly, this oil bank can be followed by a second shock wave, with nearly the same speed, which also decreases the oil saturation. Whereas the oil bank is a two-phase Buckley-Leverett fast shock wave (the saturation of one of the injected phases is almost constant), the second shock wave is of a kind only recently understood, called a transitional shock wave. Such shock waves occur in models where characteristic speeds can coincide; their structure depends sensitively on diffusion. (Physical diffusion originates from capillary pressure, but the simulation of these waves can also be affected by numerical diffusion.) Behind the transitional shock wave is the injection region, where the oil saturation decreases smoothly to zero at the injection well while the water and gas saturations oscillate. High oil recovery can result from the combined effect of the Buckley-Leverett and transitional shock waves. We discuss how the analysis of the nonlinear waves in three-phase flow can be used to devise mathematically optimal WAG strategies for oil recovery.
talk021400: Title: Flow-orientation coupling in shear flows of nematic liquid crystalline polymers Qi Wang Department of Mathematics Indiana University-Purdue University at Indianapolis Polymeric liquid crystalline materials have properties between the isotropic liquid and crystalline solid, namely, their molecules can be partially ordered under certain conditions. Therefore, flows of liquid crystalline polymers (LCP) may exhibit both viscous and elastic behavior. In this talk, we will discuss the issue of flow-orientation coupling in simple flows (shear) of liquid crystal polymers (LCP) using the extended Doi theory with Marrucci-Greco potential (DMG) for long-range molecular interaction. In this model, there are two elastic effects, one is the short range elasticity corresponding to the molecular relaxation and the other is the long range elasticity due to nonlocal molecular interaction, in addition to the viscous effects related to the viscous solvent and polymer-solvent interaction. Both the short and long range elasticity in shear flows of liquid crystalline polymers have been studied in imposed kinematics. Very little has been done for the flow-orientation coupled system however. As we will show the flow-orientation coupling is quite significant in many cases. We will begin with the construction of exact solutions of the flow-orientation coupled system, which are responsible for variety of orientational patterns. Then, we will study numerically the solution behavior of the DMG model in shear flows of LCPs with emphasis on the induced (secondary) flows due to the variety of molecular orientation patterns and try to link the orientation patterns in the flow field with the exact solution families of the DMG model.
talk030600: Special Seminar co-sponsored by the Departments of Applied Sciences, Brookhaven National Lab. Geosciences, SUNY - Stony Brook Appl. Math & Stat., SUNY - Stony Brook Mon. March 6, 4:00 PM (Refreshments at 3:45 PM) Room 123, Earth and Space Sciences Dr. Pierre Adler Institut de Physique du Globe de Paris Real porous media: geometry and transports
talk031600: Ruhai Zhou Dept. of Math & Stat Univ. of New Mexico Albuquerque, NM TIME: 11 am. Title: ----- Simulation of Unsteady Combustion Phenomena Using Complex Models Abstract: -------- A numerical method for simulating flame propagation using complex physical and chemical models is discussed. Fourth order discretization is used for the spatial derivatives. For the temporal integrations, we use preconditioning to produce a highly efficient linearly implicit method with good stability properties. Spectral deferred correction is then employed to get higher order accuracy in time. Solution adapted moving grids are used to track the flame fronts. We test our method for the hydrogen-air system using plane flame solutions as initial data. The results demonstrate the accuracy, stabilty, and efficiency of the method.
Kolmogorov Flow Yuan-nan Young, ASCI FLASH Center Department of Astronomy and Astrophysics University of Chicago In this study we investigate stratified Kolmogorov shear flow. We derive the amplitude equations for ths system and solve them numerically to explore the effect of a weak stabilizing stratification. We then explore the non-diffusive limit of this system and derive amplitude equations. This work is a collaboration with Neil Balmforth and William Young.
talk041200: Nonlinear evolution of unstable fluid interface Snezha I. Abarzhi Department of Applied Math and Statistics SUNY at Stony Brook We report solutions of the problem of the nonlinear motion of ideal fluid with a free surface and with no external forces. The motion of the free surface is associated with generation of bubbles and spikes by the Richtmyer-Meshkov instability. At a late time parameters of the regular bubble are not uniquely determined by the value of spatial period of the flow and there exists a family of regular asymptotic solutions. We make the local stability analysis for the solutions and show that bubbles with a flattened surface are faster and more stable than narrow bubbles with the radius of curvature of order of half of spatial period both in 3D and 2D.
talk050300: Tubuloglomerular Feedback-Mediated Dynamics in Two Coupled Nephrons E. Bruce Pitman Department of Mathematics 106 Diefendorf Hall State University of New York Buffalo, NY 14214-3093, U.S.A. Previously, we developed a ``minimal'' dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semilinear hyperbolic partial differential equation was used to represent two fundamental processes of mass transport in the nephron's thick ascending limb (TAL): chloride advection by fluid flow through the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate nephron glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for suffficiently large feedback gain and time delay. %perhaps from a Hopf bifurcation. In this study, the single-nephron model has been extended to two nephrons, which are coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time-delays, but differing gains, and (2) identical feedback gain magnitudes, but differing time delays. Similar to the case of a single nephron, the analysis indicates that LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, %or shorter time delays, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO.
talk052500: Interface tracking with application to multiphase flow Anna-Karin Tornberg UCLA Wavelet based methods for numerical homogenization Bjorn Engquist UCLA
talk053100: The Development of RT-Instability in a Shell of an Incompressible Fluid in the Presence of Ablation Ivan Lebo Abstract The linearized model of the RT-instability development at the surfaces of incompressible fluid shell with allowance for ablation is presented. The main ability of the ablation process to decrease the speed of RTI process, as it was shown in a wide number of earlier papers, approved in our case too. Nevertheless, our results appeared to be in qualitative and quantitative disagreement with well known Bodner-Takabe formula.
talk092700: NEW EULERIAN METHOD FOR THE COMPUTATION OF PROPAGATING SHORT WAVE EQUATION PULSES John Steinhoff The University of Tennessee Space Institute Tullahoma, TN 37388 Abstract A new method is described to compute short wave equation pulses that propagate according to geometrical optics. The pulses are treated as zero thickness sheets that can propagate over arbitrarily long distances with multiple reflections. The method has many of the advantages of Lagrangian ray tracing, but is completely Eulerian, typically using a uniform Cartesian grid. Accordingly, it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse. Also, information describing the pulses, which are treated as continuous surfaces, can be available throughout the computational grid, rather than only at isolated individual markers. The method uses a new type of representation, which we call ``Dynamic Surface Extension''. The basic idea is to propagate or ``broadcast'' defining fields from each pulse surface through a computational grid. These fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes, such as amplitude. Thus the emphasis is on the dynamics of these propagating defining fields, which obey only local Eulerian equations at each node. A description of the method together with results in 1-D, 2-D and 3-D for propagating and reflecting pulses will be presented.
talk101100: Title : A Geometric Analysis on 3D Fiber Networks from High Resolution Images Hyunmi Yang Department of Applied Mathematics and Statistics State University of New York at Stony Brook Abstract : A thorough understanding and analysis of geometry and topology of three-dimensional fiber networks from high resolution images is an important and challenging task due to the enormous complexity and randomness of the structure. In this paper we propose a technique that is aimed at structural analysis of fibrous materials. A sequence of image processing techniques is applied to the images, to obtain the medial axis of the fiber network. A description of the network is then determined from the medial axis. We demonstrate computational algorithms that can efficiently identify individual fibers from a network of randomly oriented and curled fibers that are bonded irregularly with each other. We can accurately measure the orientation, location, curl, length, bonds, and crossing angles of the identified fibers as well as the density of the fibers contained in a given volume. The performance of the proposed technique is presented for simulated fiber data and for a synthetic (nonwoven polymer) fiber data.
talk111300: The Toy Factory - Simple Models for Simple Flows Todd Dupont Department of Mathematics University of Chicago Abstract Techniques that reduce complex systems of partial differential equations to smaller more tractable forms are widely used. This talk will illustrate how semi-discretization using various numerical methods has been used to derive simple models of several interesting fluids problems.
talk111600: Experiments and Simulations of a Shock-Accelerated Gas Cylinder C.A. Zoldi in collaboration with K. Prestridge, P.M. Rightley, P. Vorobieff, R.F. Benjamin Los Alamos National Laboratory The evolution of a cylinder of SF6 gas accelerated by a Mach 1.2 shock is studied both experimentally and computationally. The experiments are performed using a horizontal shock tube. A vertical cylinder of SF6 surrounded by ambient air is impulsively accelerated by a normal Mach 1.2 shock. This produces Richtmyer-Meshkov Instability. As the cylinder evolves, the presence of Kelvin-Helmholtz Instability is also evident. Using CCD cameras and a laser sheet, images are taken of the initial conditions and the time evolution of the cylinder. Particle Image Velocimetry (PIV) is used to gather velocity measurements from the experiment. Using an image of the experimental initial conditions, 2D simulations are performed using RAGE, LANL's adaptive mesh Eulerian code. Density and velocity fields from the simulations are compared against the experimental measurements. The gross features present in the experiment are reproduced in the simulations, except that the linear dimensions are significantly smaller. The simulations also exhibit more rollup inside the vortices. This is due to the presence of higher velocities in the simulation than in the experiment. Possible reasons for the discrepancies will be discussed. This study will aid in code validation efforts and provide knowledge to help understand future experiments with more complex initial conditions.
talk020701: Luoding Zhu Courant Institute of Mathematical Sciences, New York University This talk reports the computer simulation of a flexible flapping filament in a flowing soap film using the Immersed Boundary Method. Our mathematical formulation includes the filament mass, the gravity, the air resistance, the bending force and the two wires that bound the flowing soap film. The incompressible viscous Navier-Stokes equations are discretized on a fixed uniform Eulerian lattice while the the filament equations are discretized on a moving Lagrangian array of points which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction between the filament and the soap film is handled by a smoothed approximation to the Dirac delta function. This delta function approximation is used not only to interpolate the fluid velocity and to apply force to the fluid (as is commonly done in immersed boundary computations), but also to handle the "added mass" of the filament, which is represented in our calculation as delta function layer of fluid mass density supported along the immersed filament. Because of this nonuniform density, we need to use a multigrid method for solving the discretized fluid equations. This replaces the FFT based method that is commonly used in the uniform-density case. Our main results are: 1)the sustained flapping of the filament only occurs when filament mass is included in the formulation of the model; the more mass of the filament the bigger amplitude of the flapping. 2) When the length of filament is short enough (below some critical length), the filament always approaches its straight state and when the filament length is long enough (above some critical length), it always settles into its flapping state, but when the length is between these critical values, the system is bi-stable, which means that it can settle into either state depending on the initial conditions. This numerical result agrees with that of the experiment even though the Reynolds number of the computations is lower than that of laboratory experiment by two orders of magnitude. 3)The bi-stable phenomenon occurs in a wide range of Reynolds number and it depends not only on the Reynolds number but also on some other parameters such as the non-dimensional number Rm (defined as the ratio of the filament mass and the film mass over an area of $L^2$, where $L$ is the filament length) and the filament bending rigidity.
talk022701: On the miscible Rayleigh-Taylor instability Yuan-Nan Young Northwestern University We investigate the miscible Rayleigh-Taylor (RT) instability in both 2 and 3 dimensions using direct numerical simulations, where the working fluid is assumed incompressible under the Boussinesq approximation. We first consider the case of randomly perturbed interfaces. With a variety of diagnostics, we develop a physical picture for the detailed temporal development of the mixed layer: We identify three distinct evolutionary phases in the development of the mixed layer, which can be related to detailed variations in the growth of the mixing zone. Our analysis provides an explanation for the observed differences between two and three-dimensional RT instability; the analysis also leads us to concentrate on the RT models which (1) work equally well for both laminar and turbulent flows, and (2) do not depend on turbulent scaling within the mixing layer between fluids. These candidate RT models are based on point sources within bubbles (or plumes) and interaction with each other (or the background flow). With this motivation, we examine the evolution of single plumes, and relate our numerical results (of single plumes) to a simple analytical model for plume evolution.
talk031401: New Developments in Scientific Computing James Glimm Department of Applied Math and Stat SUNY at Stony Brook A massive change in the nature of scientific computing is in progress. The change from individual investigator to big science is underway. This change will impact all of us in how we conduct computations. It will open up a number of opportunities and it will close off some of them as well. The basic driver for the change is the high cost of development of software and the slow progress in adapting to change. This leads to organization at a higher level to allow and encourage sharing of software. A further driver is the current and future availability of teraflop computers, available for scientific investigations. Thus the opportunity is easy access to high quality software, and the opening of teams developing it, with the chance to participate. A further opportunity is the openness of applications to new software developed in this manner. And the access to hardware to make use of the software for solving large scale problems of science. As part of this process, Stony Brook is participating in a proposed Center comprising five DOE laboratories and two universities to develop computational science software for adaptive gridding and higher order discretization for terascale computing. The proposed Center will interact with five major application centers, some five other application topics, and some five software centers with complementary technologies (visualization, linear equation solvers, optimization, etc.). Clearly this process, which is just now in a formative stage, will have a profound influence on the nature of computational science. Computational science as understood at Stony Brook is much closer to this model than are the programs at many univesities. Thus while we do have things to learn, and old habits to unlearn, the transition will be easier for us than for many. In particular, we have a chance to catch the train before it leaves the station.
talk032801: ANDREI V. BAEV Faculty of Computational Mathematics and Cybernetics Moscow State University Much work has been done on inverse scattering problems for one di-mensional acoustic media. Unfortunately the Gel'fand-Levitan integral equations linear method is developed only for inversion from reflection data in case of negligible dissipative effects. On the contrary, measured data, in geophysical prospecting particularly, is often influenced by these effects. The main result of this work is necessary and sufficient condition of solvability of the inverse dissipative scattering problem as data is given on a finite time interval. Also, we consider a bilinear procedure for recovering a dissipative coefficient from the reflection data. The approach is based~on~a step-by-step split method for numerically solving a system of the Gel'fand-Levitan integral equations under nonlocal boundary-value conditions. Besides, we develop a method of inverting difference schemes for solving the inverse dissipative scattering problems.
talk040401: The nonlinear motion of the interface between two fluids with different densities. S.Abarzhi We propose the first, by our knowledge, analysis of the nonlinear motion of the unstable fluid interface between two fluids with different densities (the Atwood numer is less than one). The motion of the interface is governed by the conservation laws. We derive a dynamical system describing the local dynamics of the flow and find the asymptotic solutions to the system. It is shown that the Layzer-type single-mode approximation does not conserve flux through the interface, and strictly speacking, re-scaling g -> g*A is not applicable in the nonlinear regime. We find a solution providing continuity of the normal component of the velocity through the interface. A bubble corresponding to this solution has parameters which differ sigfnificantly from those of the Layzer-type bubble. Our approach has however a number of constraints. We discuss the region of validity of the theory and its futher development.
talk040901: Maximum Principle Preserving Schemes for Elliptic Interface Problems Zhilin Li Center For Research in Scientific Computation & Mathematics North Carolina State University New finite difference methods using Cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients, singular sources, and non-smooth or even discontinuous solutions. The new finite difference schemes are constructed to satisfy the sign property of the maximum principle using quadratic optimization techniques. Convergence proofs are provided for the first and second order methods by constructing comparison functions. The methods are coupled with a multigrid solver. Numerical examples are also provided to show the efficiency of the proposed methods. I will also discuss the finite element method using structured or Cartesian grids for elliptic interface problems. Non-conforming and conforming basis functions are constructed so that the jump conditions are satisfied.
talk041801: Structural instability and ill-posedness in viscous fingering and crystal growth Saleh Tanveer Department of Mathematics Ohio-State University Abstract The displacement of a more viscous fluid by a less viscous fluid in a Hele-Shaw cell or the growth of a crystal in an undercooled melt remain areas of intensive research. Quantities of physical interest include the width of a steady viscous finger or the tip-curvature of a steady needle crystal. The stability of these solutions, as well as nonlinear time dependent features of solution including coarsening are also of interest. The literature is vast in these areas, and there have recently been a few claims that are inconsistent with previous results. A starting point for most analytical and asymptotic theories are the zero surface tension/surface energy approximation when curvature term drops out and the equations become solvable exactly in many cases. Unfortunately, however, the problem is structurally unstable at this value of surface tension meaning that the solution set for nonzero surface tension, even as surface tension tends to zero is not the same as that for zero surface tension itself. The consequence of this with respect to different types of regularizations will be discussed and it will be pointed out that a global selection principle based simply on the zero surface tension solution is not possible. We also present some recent theorems validating earlier formal asymptotics. The zero surface tension problem also forms an ill-posed problem in the sense of Hadamard for any physically reasonable norm introduced to describe the interface. As a consequence of this, small surface energy can create singular effects on the initial value problem in O(1) time. Some formal as well as rigorous results will be discussed for the intial value problem.
talk042501: Well-posedness theory for systems of hyperbolic conservation laws Taiping Liu Mathematics Department Stanford University We will discuss the joint work with Tong Yang on the well-posedness theory for conservation laws. The main ingradients are the Glimm existence theory, wave tracing method, and the entropy functional. We will explain these and also raise the open problems.
talk060401: FLOW AND TRANSPORT IN HETEROGENEOUS FORMATIONS OF A BIMODAL STRUCTURE Gedeon Dagan Faculty of Engineering Tel Aviv University, Israel Flow of uniform mean velocity $\mathbf{U}$ takes place in a heterogeneous medium made up from a matrix of conductivity $K_{0}$ and inclusions of a different conductivity $K$. The inclusions of given shape are implanted at random and independently in the medium, without overlapping. The aim of the study is to derive simple, approximate, solutions of advective transport of solutes in such heterogeneous formations, for arbitrary permeability ratio $% \kappa =K/K_{0}$ and inclusions volume fraction $n$. Transport is characterized by the spatial moments, which in turn are equal to the one particle trajectory statistical moments, for ergodic plumes. First, the dilute limit $n\ll 1$ is considered. In this case the velocity fields of different inclusions do not interact and can be determined analytically for a few shapes (circles and spheres for isotropic media and ellipses and spheroids for anisotropic ones). The trajectories moments as function of time are determined by two quadratures. The large time asymptotic results of \textit{Eames and Bush} (1999), for which trajectories have a Gaussian distribution and the longitudinal macrodispersivity becomes constant, are recovered. Next, the flow and transport problems are solved for the aforementioned simple shapes and for dense systems $n=\mathbf{0}(1)$ by using the model of composite inclusions of \textit{Hashin and Shtrikman }% (1962). The results tend to the dilute limit for $n=\mathbf{o}(1)$. Asymptotic, first-order analytical results are derived for large time, dilute systems and for weak heterogeneity ($\kappa \simeq 1$); they coincide with those of \textit{Rubin} (1995). Similarly, simple asymptotic expressions of the macrodispersivity are derived in the same case for low permeability inclusions, $\kappa =\mathbf{o}(1)$. A few illustrations of the results for dilute systems and large time macodispersivity of \textit{Eames and Bush} (1999) and \textit{Lessoff and Dagan} (2001) are briefly discussed. New results concerning the time dependent spatial moments, the higher-order statistical moments and the trajectories probability density function (\textit{Dagan and Fiori, submitted)} are presented. \textit{References} Eames, I. and J.W.\ Bush, Longitudinal dispersion by bodies fixed in a potential flow, \textit{Proc. R. Soc. Lond. A.,} 455, 3665-3686, 1999. Hashin, Z., and S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials,\textit{\ Journ. of Appl. Physics}, 33, 3125-3131, 1962. Lessoff, S.C., and G. Dagan, Solute transport in heterogeneous formations of bimodal conductivity distribution 2. Applications, \textit{Water Resour. Res., 37, 473-480, 2001.} Rubin, Y., Flow and Transport in bimodal heterogeneous formations, \textit{% Water Resour. Res.}, 31, 2461-2468, 1995.
talk112901: Vortex Phenomena and Amplitude Growth of Richtmyer-Meshkov Flows Norman Zabusky Department of Mechanical Engineering Rutgers University We examine the dynamics of a variety of geometrical configurations in planar and axisymmetric 2D and emphasize a vortex dynamical interpretation of observations and consequences. In particular for the shock accelerated : classical single mode sine wave; planar inclined interface; planar inclined curtain; cylinder and sphere. Comments will be made about resolution and accuracy and appropriate diagnostics for validation.
talk120601: Harold Trease Pacific Northwest National Laboratory An emerging DOE application for mesh-based modeling is in the area of computational biology. The potential for applying computational mesh methods spans the range from molecular interaction, protein assembly, individual cells, tissues, organs, and whole body simulations. There are identifiable computational needs in each of these areas that can benefit experiments, data analysis, and research activities. The requirements and constraints for these area are complex geometry, complex time-dependent biology/chemistry/physics, massively parallel code development, massively parallel computational resources, and cross discipline communication (experimental, computational bio-physics modeling, computer science, visualization, parallel computing, data analysis, etc.) Creating computational models that faithfully capture the geometry and physics of biological systems relies heavily on geometry generation, mesh generation algorithms, accurate discretization methods, front-tracking algorithms, and mesh quality/optimization algorithms. In this seminar I will describe the specific computational framework and application tools that we are using at the Pacific Northwest National Laboratory to model computational biology problems. Our specific applications areas are the Virtual Cell Project, the Virtual Lung Project, and the Virtual Human Project. The framework for the tools (and the tools themselves) are being developed, implemented, and deployed, in part, under the DOE SciDAC Program and the DOE Microbial Cell Project.
talk021302: Galactic Central Regions: Wavelet Methods and Numerical Simulations Chien-Chang Yen University of Minnesota Most of the nearby galaxies are found to have a central gas-dust disk. Their structures, however, are often obscured by the behind luminous star lights . We probe these structures of the galactic central regions by observation(wavelet method) and numerical simulations(relaxed method). Wavelet method decomposes a signal into various information at various levels. They are extremely useful in extracting those hidden structures of the galactic central regions. We have analyzed the NICMOS and WFPC (WFPC2) data from HST for more than 20 nearby disk galaxies. In general, the central regions are characterized by spiral or/and bar structures, and we have the following conclusions: For galaxies with a major bar, there are two possible scenarios; one is that the two-arm spirals can be traced all the way to the center; the other is a nuclear bar (bar within a bar). On the other hand, most of the galaxies without a major bar have a central or nuclear bar coupled with two-arm spirals. It is well known that spiral density waves can be generated by a rotating bar through a resonance excitation mechanism. Associated with these waves is the angular momentum transport between the bar and the disk. As waves attenuated by viscosity, the angular momentum will be deposited into the disk. This will cause the disk matter moving inward or outward, depending respectively on whether the angular momentum carried by the waves is negative or positive. Numerical simulations confirm the spiral density theory that the disk matter would gain angular momentum and move outward to form a tightly wound spiral-ring in the case of a fast bar resonance, and it would lose angular momentum and move inward to form an open-spiral and oval-ring structure in the case of a low bar. These works are supported by NSC Grant 90-2112-M-001-052.
talk022102: Date: Thursday, 2/21 Place: Math Common Room Time: Tea begin at 4:30pm, "Fermat's Last Tango" 5:00-7:00pm Fermat's Last Tango is a musical comedy, performed off broadway, and taped. We have a dvd disk of this. Play is very clever. Good to bring spouses, and significant others. Good for all levels: undergraduate, graduate, postdocs, staff and professors. All will enjoy!
talk022702: WHAT LANDSCAPE THEORY HAS TO TEACH US ABOUT SIMULATED ANNEALING Edward Weinberger Polytechnic University and Blumenthal Associates The success of simulated annealing depends critically on how configurations of high and low energy are distributed in the space of all possible solutions to the problem being considered. Evolutionary biologists, having come to the same conclusion about the importance of the locations of high and low fitness "solutions" to the problem of "optimal design" for an organism, have, by now, some useful results on how to characterize such "fitness landscapes" via a variant of Fourier analysis. A parallel development is a class of relatively simple landscapes, known collectively as "Kauffman's N-K Model", that have the useful feature that their ruggedness can be "tuned" by varying a single parameter. The goal of this talk is to explain these conceptual tools and to sketch how they might be used to improve cooling schedules, design parallel annealing algorithms, etc.
talk041702: =========================================================================== Adaptive and Parallel Discontinuous Galerkin Methods for Hyperbolic Systems Joseph E. Flaherty Scientific Computation Research Center Rensselaer Polytechnic Institute Troy, NY 12180 USA Abstract The discontinuous Galerkin method (DGM) provides an appealing approach to address problems having discontinuities, such as those that arise in hyperbolic conservation laws. Originally developed for neutron transport problems, the DGM has been used to solve both ordinary and partial differential equations. The DGM may be regarded as a way of extending finite volume methods to arbitrarily high orders of accuracy. The solution space is a piecewise continuous (polynomial) function relative to a structured or unstructured mesh. As such, it can sharply capture solution discontinuities relative to the computational mesh. It maintains local conservation on an elemental basis. Regardless of order, the DGM has a simple communication pattern to elements with a common face that makes it useful for parallel computation. It can handle problems in complex geometries to high order. And, it is useful with adaptivity since interelement continuity is neither required for h-refinement (mesh refinement and coarsening) nor p-refinement (method order variation). We describe several aspect of the method including basis construction, data structures, flux evaluation, solution limiting, local time stepping, and a posteriori error estimation. We further describe a framework for controlling parallel adaptive computation. The parallel data management system can handle high-order techniques and maintain a dynamic load balance in homogeneous and heterogeneous computing environments. Results of serial and parallel computations are are presented for unsteady compressible flow problems involving instabilities and other complex two- and three-dimensional phenomena.
talk041002: New Developments in Numerical Reservoir Simulation Zhangxin Chen Department of Mathematics Southern Methodist University This talk will address some new developments of scanning, gridding, discretizing, and visualizing technologies in numerical reservoir simulation. The scanning technology scans and extracts various geometrical data such as depth, thickness, porosity, permeability, and the location of wells, fractures, and faults. From scanning, the gridding technology generates corresponding 2D or 3D unstructured meshes. New discretization methods over these meshes have been developed. These methods are based on control volume finite elements and are capable to handle faults, horizontal wells, and unstructured meshes. The visualizing technology possesses real-time calculation and real-time display capabilities and provides streamline computations. As model examples in reservoirs, black-oil and compositional flow models will be discussed.
talk042402: Title: Designer Gene Networks: De novo constructs-in numero descriptions. Jeff Hasty Dept. of Biomedical Engineering Boston University Uncovering the structure and function of gene regulatory networks has become one of the central challenges of the post-genomic era. Theoretical models of protein-DNA feedback loops and gene regulatory networks have long been proposed, and recently, certain qualitative features of such models have been experimentally corroborated. This talk will focus on model and experimental results that demonstrate how a naturally occurring gene network can be used as a "parts List" for synthetic network design. The model formulation leads to computational and analytical approaches relevant to nonlinear dynamics and statistical physics, and the utility of such a formulation will be demonstrated through the consideration of specific design criteria for several novel genetic devices. Fluctuations originating from small molecule-number effects will be discussed in the context of model predictions, and the experimental validation of these stochastic effects underscores the importance of internal noise in gene expression. Potential biotech applications will be highlighted within the framework of cellular control schemes. Specifically, the coupling of an oscillating cellular process to a synthetic oscillator will be considered, and the resulting model behavior will be analyzed in the context of synchronization. The underlying methodology highlights the utility of engineering-based methods in the design of synthetic gene regulatory networks.
talk050102: Shock/Vortex/Entropy Interactions Gordon Erlebacher School of Computational Science & Information Technology and Department of Mathematics Florida State University I will present a series of high order numerical experiments that describe the interaction of a planar shocks with vortical and entropic structures. I will discuss the problem setup, numerical method, various types of upstream disturbances, and the structure of the shock and the downstream flow.
talk050802: The Vacuum in Isentropic Gas Dynamics Robin Young University of Massachusetts We are interested in global solutions to the equations of isentropic gas dynamics. We consider solutions having arbitrarily large data, so that the celebrated Glimm-Lax theory does not apply. One of the central difficulties in this program is the possible appearance of a vacuum. Liu and Smoller have shown that Glimm's interaction estimates do not apply near the vacuum, in that wave interactions cannot be approximated linearly. By considering interactions exactly rather than asymptotically, we analyze the vacuum in detail. It is well-known that certain Riemann problems give rise to a vacuum; we show that this is essentially the only way a vacuum can develop. We describe interactions of waves with the vacuum, and the annihilation of the vacuum. In particular, when a vacuum is annihilated, two shocks are emitted, and these form a cusp at the point of annihilation. I will describe progress on the problem of existence if time permits.
talk091602: Talk Title: Self-Similar Solutions to 2-D Riemann Problems Speaker: Prof. Suncica Canic Department of Mathematics University of Houston Abstract: In this talk a brief overview of the problems and methods used to study the structure of solutions for a class of two-dimensional Riemann problems will be presented. The speaker will focus on the analysis of models arising in gas dynamics (the steady and the unsteady transonic small disturbance equations, the nonlinear wave system) and pay a special attention on the treatment of nonlinear waves and their interaction with a nontrivial subsonic region. Since the interaction between the supersonic and subsonic flow occurs either through a transonic shock, through a rarefaction wave or via a sonic curve, different techniques need to be used to analyze the solution in each case. An overview of the techniques and a comparison between the methods used by several authors, will be given. In the end the speaker will suggest how one method can be used in the analysis of self-similar nonlinear wave structures arising in compressible Euler equations (isentropic and adiabatic case) where linearly degenerate modes are present. The corresponding reduced (self-similar) system is of mixed (elliptic-hyperbolic) type. More precisely, the density satisfied a degenerate elliptic equation, whereas vorticity satisfies a transport equation. In the low-velocity regime, the mixed system decouples (giving rise to the nonlinear wave system) and the structure of both the nonlinear and the linearly degenerate waves can be analyzed. A similarity between the structure of the decoupled systems and the fully coupled equations (corresponding to the compressible Euler equations) will be emphasized thereby hinting how the techniques presented in the first half of the lecture could be employed in the analysis of the structure of self-similar solutions of the full set of compressible Euler equations. Collaborators: Barbara Lee Keyfitz, University of Houston, Eun Heui Kim, CalState Long Beach, Gary Lieberman, Iowa State University, Dragan Mirkovic, University of Houston.
talk091902: Coupling the Sierra FEA code to smooth faceted surface evaluations in the Common Geometry Module (CGM) Timothy J. Tautges Sandia National Laboratories Albuquerque, NM, USA e-mail: tjtautg@sandia.gov Recent advances in the speed and capability of computational simulation are driving the incor- poration of geometric modeling methods in computational simulation codes. Several examples of analysis methods making use of geometric modeling include adaptive mesh refinement on curved boundaries and modeling of free surface flows over curvilinear bodies. This trend is also reflected on the pre-processing side, where mesh generation tools are forging ever-closer links to CAD tools and other sources of continuous domain representations. These efforts can all be thought of as restoring associative links between the various representations of the spatial com- putational domain. The Common Geometry Module, or CGM, is a set of libraries providing a consistent interface to geometric models in a variety of representation formats. CGM includes links to geometry in the ACIS modeling format as well as facet-based and virtual geometry representations. CGM can be linked into analysis codes to provide the same geometry functionality used in mesh gen- eration codes; in fact, the CUBIT mesh generation code accesses all its geometry functionality through CGM. We have developed a smooth facet-based surface representation in the CGM framework, where facet-based surfaces support C2-continuous differential geometry evaluations. In this presenta- tion we describe the use of facet-based surfaces to support adaptive mesh refinement in the SIERRA finite element code. Techniques used for minimizing data duplication and for associating the triangle-based facets needed by CGM to the (possibly non-conformal and h- refined) quadrilateral and triangle elements in SIERRA will be described. A general discussion of coupling physics codes to the CGM geometry component will conclude this talk. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000.
talk092502: Accurate Computation of Tidal Bores in Estuaries Professor Grafton W. H. Hui Hong Kong University of Science and Technology Tidal waves and bores belong to shallow-water flow, which is traditionally formulated in terms of water depth and fluid velocity. This formulation enjoys great success for flow with horizontal bottom and zero friction when the governing equations reduce to conservation laws. It, however, encounters difficulties in the presence of uneven bottom topography; in particular, it fails to replicate stationary flow and fails to compute tidal bores when the tide is receeding. To overcome these difficulties, we formulate the problem of shallow-water flow in terms of water level and fluid velocity. The non-homogeneous equations are solved using the fractional step method together with: (1) a Godunov-type scheme for the homogeneous conservation law equations and (2) a balanced discretisation for the source terms arising from bottom topography. The Riemann problem in this formulation is solved with an approximation equivalent to coarsening the grid for bottom topography by doubling its size locally. Our method exactly replicates the stationary flow, and accurately computes steady and unsteady flow. When applied to compute the famous tidal bores on the Qiantang River on the East Coast of China, it produces excellent agreement with field observations.
talk100202: A Nonconventional Eulerian-Lagrangian Single-Node Collocation Method for Unsteady-State Advection-Diffusion Equations Li Wu Department of Mathematics University of Rhode Island We developed a nonconventional Eulerian-Lagrangian single-node collocation method (ELSCM) with piecewise-cubic Hermite polynomials as basis functions for the numerical simulation to unsteady-state advection-diffusion transport partial differential equations. This method greatly reduces the number of unknowns in the conventional collocation method, and generates accurate numerical solutions even if very large time steps are taken. The method is relatively easy to formulate. Numerical experiments in one, two, and three-dimensional spaces are presented to show the strong potential of this method.
talk020503: Experimental and Computational Study of Fuel Injection Jet Constantine Tzanos Argonne National Laboratory Monochromatic synchrotron x-rays from the Advanced Photon Source (APS) at Argonne National Laboratory have been used to make time-resolved absorption measurements in the spray generated by a high-pressure diesel fuel injector. From these measurements, diesel fuel mass distributions, density and volume fraction have been determined as a function of time and position from the tip of the injector nozzle. The speed of the leading and trailing edges of the spray were also calculated. The measurements show that the fuel volume fraction drops off quickly as we move away from the tip of the nozzle. The front-tracking code FronTier has been used to analyze these experiments. The experimental measurements provide a basis for the validation of the code, and the validated code can be used to provide an understanding of the spray dynamics, and a quantitative description of spray breakup for the simulation of combustion in an internal combustion engine. Experimental measurements and analyses and the application of FronTier at ANL to analyze one of the APS experiment will be discussed. The potential application of FronTier for the design of an injector-based lithium thin-film-stripper generator will also be discussed.
talk042303: Dr. Folkert Tangerman Principal Scientist Photon Research Associates In Image Analysis Linear Analysis is your friend The statistical analysis of even a single large image, leads to a translation invariant image correlation function. Correlation functions tend to arise in two but usually disparate ways: 1. as Green's functions of suitable operators 2. resulting from convolution with uncorrelated random variables. These ways are associated with two different square root operations from the symmetric positive definite operator (Toeplitz) C, given by convolution with the correlation function: 1. (Cholesky) find a lower triangular matrix for which ACA'=Id, A=inverse Cholesky factorization of C. 2. (Principal Component Analysis) C=EDE', with E orthogonal, D diagonal While the second factorization is 'normal' factor analysis, the first is not only equally useful, but also more intriguing as the operator A tends to be a differential operator, appproximately translation invariant, with its coefficients fast computed. Example: In the one dimensional case: if C is N by N matrix for which C(i,j)=exp-|i-j|, A is banded (diagonal and one sub-band of opposite sign). Check this! In general A is banded dominated, as approximately explained by Szego's theory of Toeplitz matrices. We show how to extend this theory to higher dimensions (1 to 2 illustrative of the induction step), The result is a 'superfast' inverse Cholesky decomposition of C for 2d (and n-d) correlation functions.